Harmonic weak Maass forms of integral weight: a geometric approach

Candelori, Luca

doi:10.1007/s00208-014-1043-5

Cited by 12 publications

(24 citation statements)

References 17 publications

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“…This procedure is potentially ill-defined: when the space of cusp forms has dimension greater than one, it involves choices, since there are relations between Poincaré series. The question of whether weak harmonic lifts have irrational coefficients or not has been raised in [10,11,20]. Our results imply that these functions, despite appearances, are in fact of geometric, and indeed, motivic, origin.…”

Section: Weak Harmonic Lifts and Mock Modular Forms Of Integral Weightmentioning

confidence: 50%

“…Surprisingly this fact is not well known. It appeared for the first time implicitly in the work of Coleman [13] on p-adic modular forms, and later in [11,[21][22][23]. A direct proof in the case of level one was given in [6].…”

Section: Weakly Holomorphic Modular Formsmentioning

confidence: 99%

“…n+2 /D n+1 M ! −n can be interpreted as a space of modular forms of the second kind [6,11,23]. Indeed, it is canonically isomorphic to the algebraic de Rham cohomology of the moduli stack of elliptic curves with certain coefficients, and in particular, admits an action by Hecke operators.…”

Section: Real Frobeniusmentioning

confidence: 99%

“…2, we can easily compute the period matrix (2.13) in this basis. We find that for all γ ∈ , γ (2πi) 11 (z)(X − zY ) 10 The Petersson norm of , in its standard normalisation, is −2ω + ω − 2 11 (2πi) 22 = 0.00000103536205 . .…”

Section: Periodsmentioning

confidence: 99%

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A class of non-holomorphic modular forms III: real analytic cusp forms for $$\mathrm {SL}_2(\mathbb {Z})$$ SL 2 ( Z )

Brown

2018

Res Math Sci

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We define canonical real analytic versions of modular forms of integral weight for the full modular group, generalising real analytic Eisenstein series. They are harmonic Maass waveforms with poles at the cusp, whose Fourier coefficients involve periods and quasi-periods of cusp forms, which are conjecturally transcendental. In particular, we settle the question of finding explicit 'weak harmonic lifts' for every eigenform of integral weight k and level one. We show that mock modular forms of integral weight are algebro-geometric and have Fourier coefficients proportional to n 1−k (a n + ρa n ) for n = 0, where ρ is the normalised permanent of the period matrix of the corresponding motive, and a n , a n are the Fourier coefficients of a Hecke eigenform and a weakly holomorphic Hecke eigenform, respectively. More generally, this framework provides a conceptual explanation for the algebraicity of the coefficients of mock modular forms in the CM case.

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Section: Weak Harmonic Lifts and Mock Modular Forms Of Integral Weightmentioning

confidence: 50%

Section: Weakly Holomorphic Modular Formsmentioning

confidence: 99%

Section: Real Frobeniusmentioning

confidence: 99%

Section: Periodsmentioning

confidence: 99%

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A class of non-holomorphic modular forms III: real analytic cusp forms for $$\mathrm {SL}_2(\mathbb {Z})$$ SL 2 ( Z )

Brown

2018

Res Math Sci

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“…Bruinier, Rhoades, and Ono [2], and Candelori [9] proved that if a normalized newform has complex multiplication then the holomorphic part of a certain harmonic Maass form has algebraic coefficients; in particular, the coefficients of Z + E (z) are algebraic.…”

Section: Weierstrass Mock Modular Formsmentioning

confidence: 99%

Modular Forms and Weierstrass Mock Modular Forms

Clemm

2016

Mathematics

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Alfes, Griffin, Ono, and Rolen have shown that the harmonic Maass forms arising from Weierstrass ζ-functions associated to modular elliptic curves "encode" the vanishing and nonvanishing for central values and derivatives of twisted Hasse-Weil L-functions for elliptic curves. Previously, Martin and Ono proved that there are exactly five weight 2 newforms with complex multiplication that are eta-quotients. In this paper, we construct a canonical harmonic Maass form for these five curves with complex multiplication. The holomorphic part of this harmonic Maass form arises from the Weierstrass ζ-function and is referred to as the Weierstrass mock modular form. We prove that the Weierstrass mock modular form for these five curves is itself an eta-quotient or a twist of one. Using this construction, we also obtain p-adic formulas for the corresponding weight 2 newform using Atkin's U-operator.

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On coefficients of Poincaré series and single-valued periods of modular forms

Fonseca

2020

Res Math Sci

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We prove that the field generated by the Fourier coefficients of weakly holomorphic Poincaré series of a given level $$\varGamma _0(N)$$ Γ 0 ( N ) and integral weight $$k\ge 2$$ k ≥ 2 coincides with the field generated by the single-valued periods of a certain motive attached to $$\varGamma _0(N)$$ Γ 0 ( N ) . This clarifies the arithmetic nature of such Fourier coefficients and generalises previous formulas of Brown and Acres–Broadhurst giving explicit series expansions for the single-valued periods of some modular forms. Our proof is based on Bringmann–Ono’s construction of harmonic lifts of Poincaré series.

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Harmonic weak Maass forms of integral weight: a geometric approach

Cited by 12 publications

References 17 publications

A class of non-holomorphic modular forms III: real analytic cusp forms for $$\mathrm {SL}_2(\mathbb {Z})$$ SL 2 ( Z )

A class of non-holomorphic modular forms III: real analytic cusp forms for $$\mathrm {SL}_2(\mathbb {Z})$$ SL 2 ( Z )

Modular Forms and Weierstrass Mock Modular Forms

On coefficients of Poincaré series and single-valued periods of modular forms

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